[Nickj]

The oil change usually includes a filter change and will assume all of the oil is out. If you don't change the filter it is usually less oil. Also it needs to be at least warm and the bike standing vertical or quite a bit of oil remains in the engine.
The fuel in the filter, if it is just fuel then it would dry out quite well by itself but the filter would need re-oiling. If you have washed it then it does need to be really dry or the cotton in the filter material will not filter well, it will not flow air very well.
It might be that if you have too much oil in the engine some has come up into the air box, the inside of the filter box would have a thin coat of oil but that should be OK now.
There is a small chamber/box between the engine and the air filter, it is possible there will be oil in this if the engine was overfilled and would be worth checking.

Plugs.. I'd try some new ones and leave the old ones somewhere warm for a few days to dry out then clean them (some solvent would be good in case they are oily). You have a centrauto or Saudade who should have some plugs or will get you the right ones.

The battery might just be old and not up to the work involved in trying to start the bike a few times, I'm guessing you tried to start up a few times. It might be worth getting this tested when it's fully charged it might not be so good when it has big loads.

You shouldn't have hurt anything, just got bits messy that should be clean.

Check the filter is clean and dry
Charge the battery
Crank the motor with plugs out and no throttle or choke
Put filter back
Put new plugs in